ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Complex Geometry of Universal Teichm"uller Space

Discontinuity, Nonlinearity, and Complexity 9(4) (2020) 559--565 | DOI:10.5890/DNC.2020.12.009

Armen Sergeev

Steklov Mathematicval Institute, Moscow, 119991, Russian Federation

Abstract

We discuss complex geometric properties of the universal Teichm\"uller space $\mathcal T$. It is a complex Banach manifold which name is motivated by the fact that all classical Teichm\"uller spaces $T(G)$, associated with compact Riemann surfaces, are contained in $\mathcal T$ as complex subvarieties. Another important subset of $\mathcal T$ is the space $\mathcal S$ of orientation-preserving diffeomorphisms of $S^1$ considered modulo M\"obius transforms. It is a K\"ahler Frechet manifold. Our interest in $\mathcal T$ was initially motivated by its relation to string theory which we have studied earlier in a series of papers.

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